Context
A way to define the Fourier transform on $L^1$ is by finding the sets of all homomorphism $\left\{ \phi \right\}$ from $L^1(\mathbb R) \to L^1(\mathbb R)$ where in the domain the product is defined by the convolution product while in the image set the product is defined point wise. So in general you would study the equation
$$ \phi(xy) = \phi(x)\phi(y) \;\; x,y \in L^1(\mathbb R) $$
similarly we can define the fourier transform for $L^1(\mathbb R^n)$ in a very similar way.
Question
I wonder if a similar technique, namely using Banach Algebras or similar structure, can be used to define a fourier transform from, say $L^1(\mathbb{R})^n$?
My attempt was to define in $L^1(\mathbb{R})^n$ product as convolution component wise, and likewise component wise component for the image set. However by following a similar producedure to the 1 dimensional case I end up with an equation which isn't really easy to treat.
So I am asking for some references on this specific topic.